Nº 511 ISSN 0104-8910

Endogenous collateral

Aloísio Araujo José Fajardo Barbachan Mario R. Páscoa

Novembro de 2003

Endogenous Collateral

Aloisio AraujoIMPA-EPGE

Estrada Dona Castorina 110, Jardim Botanico

22460-320 Rio de Janeiro, RJ- Brazil.

Jose Fajardo BarbachanIBMEC

Av. Rio Branco 108, 12 andar,

20040-001 Rio de Janeiro, RJ- Brazil.

Mario R. Pascoa ∗

Universidade Nova de Lisboa, Faculdade de Economia

Travessa Estevao Pinto, 1099-032,

Lisbon, Portugal.

November 4, 2003

JEL Classification: D52

∗We are indebted to F. Martins da Rocha and J.P. Torres-Martinez for valuablecomments and pointing out errors in an earlier version. Pascoa benefited from projectPOCTI/36525/ECO/2000 (FCT and FEDER). The Research of A. Araujo was supportedin part by FAPERJ, Brazil. A. Araujo and J. Fajardo also acknowledge financial supportfrom CNPq, Brazil. J. Fajardo also wants to thank IMPA-Brazil, where this project waspartially developed.

1

Summary

We study an economy where there are two types of assets. Consumers’promises are the primitive defaultable assets secured by collateral chosen bythe consumers themselves. The purchase of these personalized assets by fi-nancial intermediaries is financed by selling back derivatives to consumers.We show that nonarbitrage prices of primitive assets are strict submartin-gales, whereas nonarbitrage prices of derivatives are supermartingales. Nextwe establish existence of equilibrium, without imposing bounds on short sales.The nonconvexity of the budget set is overcome by considering a continuumof agents.

Keywords: Endogenous Collateral; Non Arbitrage.

2

1 Introduction

1.1 Motivation

Housing mortgages stand out as the most clear and most common caseof collateralized loans. In the past, these mortgages were entirely financedby commercial banks who had to face a serious adverse selection problem inaddition of the risks associated with concentrating investments in the hous-ing sector. More recently, banks have managed to pass these risks to otherinvestors. The collateralized mortgage obligations (C.M.O.) developped inthe eighties and nineties are an example of a mechanism of spreading risksof investing in the housing market. These obligations are derivatives backedby a big pool of mortgages which was split into different contingent flows.

Collateralized loans were first addressed in a general equilibrium settingby Dubey, Geanakoplos and Zame [9]. Collateral was modelled by these au-thors as a bundle of durable goods, purchased by a borrower at the timeassets are sold and surrendered to the creditor in case of default. Clearly, inthe absence of other default penalties, in each state of nature, a debtor willhonor this commitments only when the debt does not exceed the value of thecollateral. Similarly, each creditor should expect to receive the minimum be-tween his claim and the value of the collateral. This pionnering work studieda two-period incomplete markets model with default and exogenous collat-eral coefficients and discussed also the endogenization of these coefficients,allowing for some coefficients to prevail in equilibrium, out of a possible finiteset of strictly positive values, but for a fixed composition in terms of durablegoods. Araujo, Pascoa and Torres-Martinez [5] extended the exogenous col-lateral model to infinite horizon economies with one-period assets and showedthat Ponzi schemes can be avoided without imposing transversality or debtconstraints.

Araujo, Orrillo and Pascoa [3] studied existence of equilibria in an econ-omy where borrowers may choose collateral bundles under the restrictionthat the value of the collateral, per unit of asset and at the time when it isconstituted, must exceed the asset price by some arbitrarily small amountexogenously fixed. Under this requirement the loan can only finance up tosome certain fraction of the value of the house. Lenders were assumed not

3

to trade directly with individual borrowers, but rather to buy obligationsbacked by a weighted average of the collaterals chosen by individual borrow-ers, with the individual sales serving as weights. Borrowers sell at differentprices depending on the collateral choice, as there is a spread which is a dis-counted expectation of default given in the future. Hence, borrowers choosethe composition of the collateral in terms of durable goods and the collateralmargin (which is not necessarily equal to the exogenous lower bound as morecollateral reduces the spread).

However, the model suffered from three important drawbacks that we tryto overcome in the current paper. First, short sales were bounded due tothe above exogenous lower bound on the difference between the value of thecollateral and the asset price ( in fact, first period budget feasibility impliesthat short sales must be bounded by the upper bound on endowments di-vided by the exogenously fixed lower bound on the difference between thevalue of the collateral and the asset price). It is hard to accept the existenceof an exogenous uniform upper bound on the fraction of the value the housethat can be financed by a loan.

Secondly, the payoffs of the derivative were constructed in a way that im-plied that in equilibrium, in each state of nature, either all borrowers wouldhonor their debts or all borrowers would default (even though the collateralbundle might vary across borrower). In fact, derivative’s payoffs were as-sumed to be the minimum between the debt and the value of the depreciatedweighted average of all collateral bundles. If we require, as we do in thispaper, that the derivative’s payoff in each state is just the weighted averageof borrowers’ repayments (which may be the full repayment of the debt forsome borrowers or the value of the depreciated personalized collateral forothers), then, in equilibrium, some borrowers may default while others willpay back their loans.

Third, derivative aggregate purchases were required to match, in units,aggregate short-sales of primitive assets, but this equality should only berequired in value. That is, each financial intermediary should be financing thepurchase of the consumers’ promises on a certain primitive asset by issuingthe respective derivative, thereby making zero profit at the initial date (andalso at any future state of nature due to the above requirement that the

4

derivative’s endogenous payoff should be the weighted average of consumers’effective repayments).

1.2 Results and Methodology

It is well known that in incomplete markets with real assets equilibriummight not exist without the presence of a bounded short sales condition (seeHart [14] for a counter-example and Duffie and Shafer [10] on generic exis-tence). In a model with exogenous collateral this bounded short sales condi-tion does not need to be imposed arbitrarily but it follows from the fact thatcollateral must be constituted at the exogenously given coefficients. An im-portant question is whether existence of equilibria may dispense any boundedshort sales conditions in a model with endogenous collateral. Presumably,the fact that the borrower holds and consumes the collateral may discouragehim from choosing the collateral so low that default would become a sureevent. We try to explore this fact to show that, in fact, defaulting in everystate is incompatible with the necessary first order conditions governing theoptimal choice of the collateral coefficients. From here we derive an argumentestablishing that equilibrium levels of the collateral coefficients are boundedaway from zero and, therefore, equilibrium aggregate short sales are bounded.

Allowing borrowers to choose their collateral bundles introduces a non-convexity in the budget set, which is overcome by considering a continuumof agents. This large agents set is actually a nice set up both for the hugepooling of individual mortgages and for the spreading of risks across manyinvestors. However, for a continuum of agents, having established that ag-gregate short sales are endogenously bounded does not imply that the shortsales allocation is uniformly bounded. To handle this difficulty we appeal tothe differentiability and strict concavity of the utility function. As collateralcoefficients were already shown to be bounded from below (uniformly acrossagents), if short sales were not uniformly bounded, then the collateral bundlewould become arbitrarily large, for some sequence of borrowers, and the re-spective marginal utility of consumption would tend to zero. It is shown thatthis would contradict first order conditions. Then, short sales allocations areendogenously uniformly bounded, as desired to prove existence using a multi-dimensional version of Fatou’s lemma applied to a sequence of equilibria oftruncated auxiliary economies whose bundles and portfolios are bounded.

5

1.3 Arbitrage and Pricing

The existence argument uses a pricing formula suggested by a study ofthe nonarbitrage conditions for asset pricing in the context of a model wherepurchases of the collateralized derivatives and sales of individual assets yielddifferent returns. This nonarbitrage analysis was absent in the earlier workby [3], where budget feasible short sales were bounded.

Our analysis of the nonarbitrage conditions is close to the study made byJouini and Kallal [16] in the presence of short sales constraints. In fact, theindividual promises of homeowners are assets that can not be bought by theseagents and the collateralized derivatives bought by investors is an asset thatcan not be short sold by these agents. These sign constraints determine thatpurchase prices of the the collateralized derivatives follow supermartingales,whereas sale prices of homeowners promises follow submartingales. Actually,the latter must be strict submartingale when collateral is consumed by bor-rowers, since short sales generate utility returns also, and in this respect, ouranalysis differs from [16].

The nonarbitrage conditions identify several components in the price ofa consumer’s promise: a base price common to all consumers, a spread thatdepends on the future default, a positive term reflecting the difference be-tween current and future collateral values, a nonnegative tail due to the signconstraints and a negative tail on the sale price due to utility returns fromconsumption of the collateral. We also show that the price of the minimalcost superhedging strategy is the supremum over all discounted expectationsof the claim, with respect to every underlying probability measure (and sim-ilarly, the price of a maximal revenue subhedging strategy is instead the in-fimum over those expectations, in the spirit of the Cvitanic and Karatzas [7]and El Karoui and Quenez [12] approaches to pricing in incomplete markets).

In equilibrium agents will face price functions, as in [3], rather than pricevectors. More precisely, we propose price formulas both for the primitiveassets and the derivatives which are suggested by our arbitrage analysis. Thestate prices entering in these equilibrium price functions and the negative tailof the primitive asset prices are both taken as given and common to all agents.That is, equilibrium prices of derivative or primitive assets are given by super

6

or sub martingales, respectively, with respect to a common measure, but canalso be written as super or sub martingales for consumer specific measuresimplied by the personal choice of collateral and effective returns (namelyusing the Kuhn-Tucker multipliers as deflators).

1.4 Relation to Other Equilibrium Concepts

We close the paper with a discussion of the efficiency properties of equi-libria. We show that an equilibrium allocation is undominated by alloca-tions that are feasible and provide income across states through the samegiven equilibrium spot prices, although may be financed in the first periodin any other way (possibly through transfers across individuals). This re-sults extends usual constrained efficiency results to the case of default andendogenous collateral. An implication is that the no-default equilibrium, theexogenous collateral equilibrium or even the endogenous collateral equilib-rium with bounded short sales are concepts imposing further restrictions onthe welfare problem and should be expected to be dominated by the proposedequilibrium concept.

In this paper we simplify the mixing of individual promises by assum-ing that each collateralized derivative mixes the promises of all sellers ofa certain primitive asset. Since the collateral choice personalizes the assetthe resulting derivative represents already a significative mixing across as-sets with rather different default profiles. Further work should address thecomposition of derivatives from different primitive assets and certain chosensubsets of debtors. We do not deal also with the case of default penaltiesentering the utility function and the resulting adverse selection problems.The penalty model was extensively studied by Dubey, Geanakoplos and Shu-bik [8], extended to a continuum of states and infinite horizon by Araujo,Monteiro and Pascoa [1, 2] and combined with the collateral model by [9].Our default model differs also from the bankruptcy models where agents donot honor their debts only when they have no means to pay them, or moreprecisely, when the entire financial debt exceeds the value of the endowmentsthat creditors are entitled to confiscate (see Araujo and Pascoa [4]).

The paper is organized as follows. Section 2 presents the basic modelof default and collateral choice. Sections 3 and 4 address arbitrage and

7

pricing. Section 5 presents the definition of equilibrium and the existenceresult. Section 6 contains the existence proof and Section 7 discusses theefficiency properties. A mathematical appendix contains some results usedin the existence proof.

2 Model of Default and Collateral Choice

We consider an economy with two periods and a finite number S of statesof nature in the second period. There are L physical durable commoditiestraded in the market and J real assets that are traded in the initial periodand yield returns in the second period. These returns are represented by arandom variable R : S 7→ IRJL such that the returns from each asset are nottrivially zero. In this economy each sale of asset j (promise) must be backedby collateral. This collateral will consist of goods that depreciate at somerate Ys depending on the state of nature s ∈ S that occurs in the secondperiod.

Each seller of assets chooses also the collateral coefficient for the differentassets that he sells and we suppose that the mean collateral coefficients canbe known by consumers. For each asset j denote by Mj ∈ IRL

+ the choiceof collateral coefficients. The mean collateral coefficients will be denoted byC ∈ IRJL

+ . Each agent in the economy is a small investor whose portfolio is(θ, ϕ) ∈ IRJ

+× IRJ+ , where the first and second components are the purchase

of the derivative and sale of the primitive assets, respectively. The collateralbundle choosen by borrower will be Mϕ and his whole first period consump-tion bundle is xo + Mϕ.

Denote by xs ∈ IRl+ the consumption vector in state of nature s. Agent´s

endowments are denoted by ω ∈ IR(S+1)L++ . Let π1 and π2 be the vectors

of purchase prices of the derivatives and of sale prices of primitive assets,respectively. Then, the budget constraints of each agent will be the following

poxo + poMϕ + π1θ ≤ poωo + π2ϕ (1)

psxs +J∑

j=1

Dsjϕj ≤ psωs +J∑

j=1

Nsjθj +J∑

j=1

psYsMjϕj + psYsxo, ∀s ∈ S (2)

8

Here Dsj ≡ minpsRjs, psYsMj and Nsj are what he will paid and received

with the sale and purchase of one unit of the primitive asset j and one unitof its derivative, respectively. Now we will represent equations (1) and (2) inmatrix form:

p 2(x− ω) ≤ A(xo, θ, ϕ) (3)

where x = (0, x1, . . . , xS), ω = (ωo, ω1, . . . , ωS), p 2(x − ω) is the columnvector whose components are ps · (xs − ωs) for s = 0, 1, . . . , S and

A =

−po −π1 π2 − poMp1Y1 N1 p1Y1M −D1

p2Y2 N2 p2Y2M −D2

· · ·· · ·pSYS NS pSYSM −DS

3 Arbitrage and Collateral

Now we will define arbitrage in our context where both sales of collateral-ized assets and additional purchases of durable goods have utility returns thathave to be taken into account together with pecuniary returns. Moreover,agents’ preferences are assumed to be monotonic.

Definition 1 We say that there exist arbitrage opportunities if∃ Mj > 0, j = 1, .., J, θ ≥ 0 and (xo, ϕ) such that

T (xo, θ, ϕ) > 0 (4)

where

T =

AI 0 00 0 I

Notice that even when there are no pecuniary net returns and zero net costthe agent may still gain from the utility returns of consuming durable goods,serving or not as collateral, that is, through a collateralized short sale (ϕj >0) or a non financed purchase (xo > 0).

9

Theorem 1 There are no arbitrage opportunities if and only if there existsβ ∈ IRS

++ such that for each j = 1, 2, .., J

πj1 ≥

S∑s=1

βspsNjs (5)

πj2 <

S∑s=1

βspsRjs −

S∑s=1

βs(psRjs − psYsMj)

+ + (poMj −S∑

s=1

βspsYsMj) (6)

and

po >

S∑s=1

βspsYs (7)

Proof:Let B = T (xo, θ, ϕ) : θ ≥ 0 and B = T (xo, θ, ϕ) : θ = 0, which are aconvex cone and a linear subspace, respectively. Let K = IR+× IRS

+× IRL+J+ .

Absence arbitrage is equivalent to K ∩B = 0. By the theorem of sep-aration of convex cones, we have that K ∩ B = 0 if and only if ∃f 6= 0linear: f(z) < f(y), ∀z ∈ B, y ∈ K\0.

Now f(z) = 0, ∀z ∈ B, since B is a linear subspace. Then f(y) >0, ∀y ∈ K\0 and it follows that f(z) ≤ 0 ∀z ∈ B. Hence ∃ (α, β, µ, η) >>0 : f(v, c, xo, ϕ) = α + βc + µxo + ηϕ ≤ 0, ∀(v, c, xo, ϕ) ∈ B. Takeβ = β/α, µ = µ/α and η = η/α, and we have (5) when (xo, ϕ) = 0. Toobtain (6) and (7) let θ = 0 and recall that f(z) = 0, ∀z ∈ B, implying

poMj − πj2 =

∑s

βs(psYsMj −Dsj) + ηj

andpo =

∑s

βspsYs + µ. ¥

Comment

Durable goods prices (p0) and net prices (p0Mj−πj

2) of the joint operation ofconstituting collateral and short-selling a primitive asset are both superlinearfunctions of pecuniary returns, by the Theorem above, due to the additionalutility returns from consumption (of x0 and of M jϕj, respectively).

10

Corollary 1

poMj − πj2 > 0 when Mj 6= 0, ∀j

Since short-sales lead to nonnegative net yields in the second period (oncewe add the depreciated collateral to returns) and also to consumption of thecollateral bundle in the first period, nonarbitrage requires the net coefficientof short-sales in the first period budget constraint to be positive.

If we had considered the collateral as being exogenous, we would havethe following result:

Corollary 2 There are no arbitrage opportunities if and only if there existsβ ∈ IR

(S)++ such that

S∑s=1

βsDsj ≤ πj <

S∑s=1

βsDsj + (po −S∑

s=1

βspsYs)Cj

, which implies

(po −S∑

s=1

βspsYs)Cj > 0, and poCj − πj > 0, ∀j ∈ J.

For more details on the implications of the absence of arbitrage in theexogenous collateral model see Fajardo [13].

In contrast with the fundamental theorem of asset pricing in frictionlessfinancial markets, we can obtain an alternative result for the default modelwith collateral where discounted nonarbitrage asset prices are no longer mar-tingales with respect to some equivalent probability measure. This result ispresented in the next section.

4 Pricing

4.1 A Pricing Theorem

Let IR be the real line and IR = IR ∪ −∞, +∞ the extended real line.Let Ω = 1, 2, .., S, (Ω,F , P ) be a probability space and X = IRS. We say

11

that f : X 7→ IR is a positive linear functional if ∀ x ∈ X+, f(x) > 0, whereX+ = x ∈ X/P (x ≥ 0) = 1 and P (x > 0) > 0. The next result followsin spirit of the result in [16].

Let πj2 = πj

2 − poMj < 0, ∀j which will be refered to as the net sell priceand let Dsj = Dsj − psYsMj, ∀j and ∀s.

Denote by ι(x) the smallest amount necessary to get at least the payoffx for sure by trading in the underlying defaultable assets. Then no investoris willing to pay more than ι(x) for the contingent claim x. The specificexpression for ι is given by

ι(x) = inf(θ,ϕ)∈Θ

π1θ − π2ϕ > 0/

G(θ, ϕ) ≥ x a.s.

where

G(θ, ϕ) =J∑

j=1

[Njθj −Djϕ

j]

Theorem 2 i) There are no arbitrage opportunities if and only if thereexist probabilities β∗s , s = 1, .., S equivalent to P and a positive γ suchthat the normalized (by γ) purchase prices of the derivatives are super-martingales and the normalized (by γ) net sale prices of the primitiveassets are submartingales under this probability. when the collateral isconsumed by the borrower, the net sale price is a strict submartingale

ii) Let Q∗ be the set of β∗ obtained in (i) and Γ be the set of positive linearfunctionals ξ such that ξ|M ≤ ι, where M is a convex cone representingthe set of marketed claims. Then there is a one-to-one correspondencebetween these functionals and the equivalent probability measures β∗

given by:

β∗(B) =S∑

s=1

β∗s1B(s) = ξ(1B) and ξ(x) = E∗(x

γ)

where E∗ is the expectation taken with respect to β∗

iii) For all x ∈M we have

[−ι(−x), ι(x)] = clE∗(x

γ) : β∗ ∈ Q∗

12

Proof:

(i) Let βo =∑S

s=1 βs and β∗s = βs

βoin theorem 1, we obtain:

πj1/βo ≥

S∑s=1

β∗sNsj

and

πj2/βo ≤

S∑s=1

β∗spsRjs −

S∑s=1

β∗s (psRjs − psYsMj)

+

+(poMj/βo −S∑

s=1

β∗spsYsMj)

Take γ = 1/βo. From the above equations it follows that πj1 and

πj2 − p0Mj are super and sub martingales, respectively.

Now, if there is a probability measure and a process γ such the nor-malized prices are sub and supermartingales, we have

E∗(

J∑j=1

[N jθj −Djϕj]

)≤ γ[π1θ − π2ϕ]

Then there can not exists arbitrage opportunities.

(ii) Given β∗ ∈ Q∗ define ξ(x) = E∗(xγ), then

ξ(x) =∑

s

(xs

γ

β∗sPs

)

it is a continuous linear functional. Since β∗ is equivalent to P andtaking the infimum over all supereplicating strategies :

E∗(x) ≤ E∗(

J∑j=1

[N jθj −Djϕj]

)≤ γ[π1θ − π2ϕ]

we have ξ ∈ Γ.Now take ξ ∈ Γ and define β∗(B) =

∑Ss=1 β∗s1B(s) = ξ(1B). Since S is

13

finite, β∗ is equivalent to P .

Now since ξ(1S) = 1, we have β∗(S) = 1 =∑S

s=1 β∗s , so β∗ is a proba-bility.

(iii) By part (ii) take a ξ ∈ Γ then ∀x ∈Mξ(x) ≤ ι(x) ⇒ −ξ(−x) ≤ ι(x)

then replacing x by −x we have

ξ(x) ≥ −ι(−x)

Hence

clξ(x)/ξ ∈ Γ ⊂ [−ι(−x), ι(x)]

For the converse, −ι(−x) = ι(x) the proof is trivial. Then we supposethat −ι(−x) < ι(x). Now it is easy to see that ι is l.s.c. and sublinear.Then the set K = (x, λ) ∈ M × IR : λ ≥ ι(x) is a closed convexcone. Hence ∀ε > 0 we have that (x, ι(x) − ε) /∈ K. Applying thestrict separation theorem we obtain that there exist a vector φ andthere exists real number α such thatφ · (x, ι(x)− ε) < α and φ · (x, λ) >α ∀(x, λ) ∈ K. Then we can rewrite these inequalities as:

φo · x + φS+1(ι(x)− ε) < α

φo · x + φS+1λ > α ∀(x, λ) ∈ K

where φo = (φ1, ..., φS) and, since K is a convex cone, we must haveα < 0. This implies φo · x + φS+1(ι(x) − ε) < 0 and φo · x + φS+1λ ≥0 ∀(x, λ) ∈ K. Hence φS+1 > 0 and we can define ν(x) = − φo

φS+1· x.

It is easy to see that ν is a continuous linear functional and ν(x) ≤ι(x), ∀x ∈ M, since (x, ι(x)) ∈ K. Also ν(x) > ι(x) − ε. Now for allx ∈ X+, we have ν(−x) ≤ ι(−x) ≤ 0, so ν(x) ≥ 0. With an analoguousargument, we obtain ν ′(x) ∈ Γ such that ν ′|M ≤ ι and

−ι(−x) ≤ ν ′(x) ≤ −ι(−x) + ε

Since ν ∈ Ξ/ν|M ≤ ι is a convex set and ν(x)/ν|M ≤ ι , ν ∈ Γ isan interval we obtain the inclusion.¥

14

Remark

• Our definition of maximal willingness to pay ι(x) is in the spirit ofthe super replication approach of [12] and [7] to pricing in incompletemarkets. We consider as superhedging strategies the defaultable assets.

Theorem 2, (ii) establishes a one to one correspondence between linearpricing rules, bounded from above by ι(x), and measures β∗, consideredin the sub and supermartingale pricing formulasOur result (iii) implies

[ infβ∗∈Q∗

E∗(x

γ), sup

β∗∈Q∗E∗(

x

γ)] = [−ι(−x), ι(x)]

5 Equilibria

In this section borrowers (sellers of assets) will choose the collateral coeffi-cients. We assume that there is a continuum of agents H = [0, 1] modeled bythe Lebesgue probability space (H,B, λ). Each agent h is characterized byhis endowments ωh and his utility Uh. Each agent sells in the initial periodJ assets that will be backed by a chosen collateral bundle and purchases alsothe derivatives; in the second period will receive the respective returns.

The allocation of the commodities is an integrable map x : H → IR(S+1)L+ .

The derivative purchase and primitive assets short sale allocations are rep-resented by two integral maps; θ : H → IRJ

+ and ϕ : H → IRJ+, respectively.

Each borrower h will choose the collateral coefficients for each portfolio sold.The allocation of collateral coefficients chosen by borrowers is described bythe function M : H → IRJ

+.

Consumers short-sell and collateralize the primitive assets but can onlybuy a derivative issued by a financial intermediary that buys the primitiveassets. The value of the derivative’s aggregate purchases must match thevalue of the primitive asset’s aggregate short-sales (and the value of the ag-gregate respective returns should also be equal in any state of nature in thefuture). Each buyer of assets (lender) will take as given the derivatives’ pay-offs Nsj and a mean collateral coefficients vector C ∈ IRJL

+ as given. Let

15

xh−o = (xh

1 , . . . , xhS) be the commodity consumption in the several states of

the world in the second period.

Sale prices of primitive assets are assumed to consist of a base priceminus a discounted expected value of future default plus a term reflecting thecollateral requirements (which entail a cost but yield a depreciated collateralbundle) and an addicional negative tail δj ≡ −(p0 −

∑s γspsYs)Cj which is

independent of the collateral choice. More specifically we assume

π2j = qj −∑

s

γs(psRsj − psYsMj)+ + (po −

∑s

γspsYs)(Mj − Cj) (8)

The state prices γs are common to all agents and taken as given togetherwith the base price q. The vector of prices for the collateralized derivatives,whose returns are given by Ns, is π1j. We will show that for an asset j whichis traded we have qj =

∑s psRsj and the price of the respective derivative

π1j =∑

s γsNsj.

Then the individual problem is

max(xh,θh,ϕh,Mh)∈Bh

Uh(xho + Mhϕh, xh

−o) (9)

where Bh is the budget set of each agent h ∈ H given by:

Bh(p, π1, q, γ, C, N) =(x, θ, ϕ, M) ∈ IRL(S+1)+2J+JL : (1) and (2) hold for

π2 given by (8)

Definition 2 An equilibrium is a vector ((p, π1, π2, C, N), (xh, θh, ϕh,Mh)h∈H)such that:

•(xh, θh, ϕh,Mh)

solves problem (9)

• ∫

H

(xh

o +∑j∈J

Mhj ϕh

j

)dh =

∫

H

ωho dh (10)

16

∫

H

xh(s)dh =

∫

H

(ωh(s) +

∑j∈J

(YsMhj ϕh

j + Ysxho)

)dh (11)

• ∫

H

Mhj ϕh

j dh = Cj

∫

H

ϕhj dh ∀ j ∈ J (12)

•Nsj

∫

H

θhj dh =

∫

H

Dsjϕhj dh, ∀j ∈ J, ∀s ∈ S (13)

•πj

1

∫

H

θhj dh =

∫

H

πjh2 ϕh

j dh. (14)

Some Remarks

• Equations (10) and (11) are the usual market clearing conditions. Equa-tion (12) says that in equilibrium the anonymous collateral coefficientCj is anticipated as the weighted average of the collateral coefficientsallocation Mj.

• Equation (13) says that aggregate yields of each derivative must beequal to aggregate actual payments of the underlying primitive assets.This implies that aggregate default suffered must be equal to aggregatedefault given, for each state and each promise:

∫

h∈Sjs

(psRjs−Nsj)

+θhj dh =

∫

h∈Gjs

(psRjs−psYsM

hj )+ϕh

j dh ∀s ∈ S, ∀j ∈ J

Where Sjs = h ∈ H : psR

js > Nsj is the set of agents that suffered

default in state of nature s on asset j and Gjs = h ∈ H : psR

js >

psYsMhj is the set of agents that give default in state of nature s on

asset j. Note that Sjs is equal to H or φ, since psR

js and Nsj do not

depend on h.

• The above equilibrium concept portraits equilibria in housing mort-gages markets where individual mortgages are backed by houses and

17

then huge pools of mortgages are split into derivatives.

In our anonymous and abstract setting, any agent in the economy maybe simultaneously a homeowner and an investor buying a derivative.The above equilibrium concept assumes the existence of J financialinstitutions, each one buying the pool of mortgages, written on prim-itive asset j, from consumers at prices πh

2j and issuing the respectivederivative, which is sold to consumers at prices π1j. These financialinstitutions make zero profits in equilibrium both at the initial dateand at any future state of nature.

To simplify, we mix promises of different sellers of a same asset butdo not mix different assets into derivatives. This simplification is nottoo strong, since different sellers of a same asset end up selling person-alized assets due to different choices of collateral. A more elaboratemodel should allow for the mix of different primitive assets and forthe strategic choice of the mix of assets and debtors by the issuer ofthe derivative. Putting together in a same model the price-taking con-sumers and investments banks composing the derivatives strategicallymay be a difficult task, since the latter would have to anticipate theWalrasian response of the former.

We will now fix our assumptions on preferences.

Assumption (P) : preferences are time and state separable, monotonic, rep-resentable by smooth strictly concave utility functions uh such that the map∇uh : H × IR

(S+1)L+ → IR is jointly continuous and ∇uh(x) → 0 uniformly

in h as ‖x‖ → ∞.

Theorem 3 If consumers’s preferences satisfy assumption (P) and the en-

dowments allocation ω belongs to L∞(H, IR(S+1)L++ ), then, there exist equilibria

where borrowers choose their respective collateral coefficients.

6 Proof of the Existence Theorem

Let us first address the case where bundles and portfolios are boundedfrom above. More precisely, nonfinanced consumption bundles xh, portfolios

18

(θh, ϕh) and collateral coefficients Mhj are bounded by n in each coordinate.

Then we will let n go to ∞.

Truncated Economy

Define a sequence of truncated economies (En)n such that the budget setof each agent h is

Bhn(p, π1, q, γ, C, N) := (xh

n, θhn, ϕh

n,Mhn ) ∈ [0, n]L(S+1)+2J+JL : (1) and (2) hold

We assume that C ∈ [0, n]LJ .

Generalized Game

For each n ∈ N we define the following generalized game played by thecontinuum of consumers and some additional atomic players. Denote thisgame by Jn which is described as follows:

• Each consumer h ∈ H maximizes Uh in the constrained strategy setBh

n(p, q, π1, C, γ).

• The auctioneer of the first period chooses po ∈ 4L−1 in order to maxi-mize

po

∫

H

(xho +

∑j

Mhj ϕh

j − ωho )dh

• The auctioneer of state s of the second period chooses ps ∈ 4L−1 inorder to maximize

ps

∫

H

(xhs − Ys(

∑j

Mhj ϕh

j + xho)− ωh

s )dh.

• The first JL fictitious agents chooses Cjl ∈ [0, n] in order to minimize

(Cjl

∫

H

ϕhj dh−

∫

H

Mhjlϕ

hj dh

)2

.

19

• Another fictitious agent chooses π1j ∈ [0, 1] , qj ∈ [0, γ maxs,k Rsjk] ,Nsj ∈ [0, n] and γs ∈ [0, γ] for every j and s in order to minimize

∑j

((π1j

∫

H

θhj dh−

∫

H

πh2jϕ

hj dh)2 + (qj −

∑s

γspsRsj)2

∫

H

θhj dh

+∑

s

(Nsj

∫

H

θhj dh−

∫

H

minpsRsj, psYsM

hj

ϕh

j dh)2

)

This game has an equilibrium in mixed strategies (see lemma 8) and, byLiapunov’s Theorem (see lemma 9), there exists a pure strategies equilibrium.

Now let us define a free disposal equilibrium for the truncated econ-omy as a pair consisting of a price vector (p, π1, γ, C, N) and an allocation(x, θ, ϕ, M)H) such that (x, θ, ϕ, M)(h) maximizes consumer h’s utility Uh onthe constrained budget set of the truncated economy given the price vectorand

∫H

(xh0 + Mhϕh − ωh

0 )dh = 0

∫H

(xhs − ωh

s − Ysxh0 − YsM

hϕh)dh ≤ 0

Nsj

∫H

θhj dh ≤ ∫

HDh

sjϕhj dh

πj1

∫H

θhj dh =

∫H

πjh2 ϕh

j dh

Cj

∫H

ϕhj dh =

∫H

Mhj ϕh

j dh

∫H

ϕhj dh = 0 ⇐⇒ ∫

Hθh

j dh = 0

Lemma 1 For n large enough, there exists a free-disposal equilibrium for thetruncated economy.

20

Proof:Let z = (xh, θh, ϕh,Mh) : H → [0, n]L(S+1)+2J+LJ , (po, q, π1,γ, ps, C) be

an equilibrium in pure strategies for Jn. Now, Cj

∫H

ϕhj dh =

∫H

Mhj ϕh

j dh.In fact, the equality holds trivially when

∫H

ϕhdh = 0 and, otherwise, noticethat

∫H

Mhj ϕh

j dh/∫

Hϕh

j dh ≤ n and therefore Cj can be chosen in [0, n] tomake this equality hold.

Claim: (1) π1

∫H

θhdh =∫

Hπh

2ϕhdh, (2) qj =∑

s γspsRsj when∫H

θhj dh 6= 0, (3)

∫H

θhdh = 0 iff∫

Hϕhdh = 0.

In fact, if∫

Hθh

j dh 6= 0 the financial intermediary chooses π1j ∈ [0, 1] and

γ ∈ [0, γ]S so that qj =∑

s γspsRsj and

π1j

∫

H

θhj dh =

∫

H

πh2jϕ

hj dh

=∑

s

γs

∫

H

Dhj ϕh

j dh

If ϕhj = 0 for a.e. h but

∫H

θhj dh 6= 0, then Nj and π1jare set equal to

zero, implying that θhj could be instead set equal to zero, for a.e. h, without

affecting any of the strategic equilibrium conditions.

If∫

Hθh

j dh = 0 the financial intermediary sets γ so that p0 ≥∑

s γspsYs

and makes qj =∑

s γs(psRsj − psYsMj)+ implying

πh2j = (p0 −

∑s

γspsYs)(Mj − Cj).

When p0 =∑

s γspsYs all borrowers choose ϕhj = 0 and when p0 >

∑s γspsYs

all borrowers choose Mhj ≥ Cj and Cj

∫H

ϕhdh =∫

HMhϕh

j dh implies Mhj =

Cj. Then, πh2j = 0 and ϕh

j = 0, ∀h.¤.

Claim: (1) Nsj

∫H

θhj dh ≤ ∫

Hmin

psRsj, psYsM

hj

ϕh

j dh, ∀s and(2) π1j ≥

∑s γsNsj

In fact, these inequalities hold as equalities when∫

Hθh

j dh = 0 (as seenabove) or when

∫H

Dhsjϕ

hj dh/

∫H

θhj dh does not exceed n, for every s. Other-

wise, the strict inequalities hold in (1) for some s and in (2).¤

21

Now, the optimality conditions of the auctioneers’ problems imply that

∫

H

(xho − ωh

o + Mhϕh)dh ≤ 0 (15)

∫

H

(xhs − ωh

s − YsMhϕh − Ysx

ho)dh ≤ 0 (16)

After integrating the budget constraint of the second period, we obtain

ps

∫

H

(xhs − ωh

s − YsMhϕh − Ysx

ho)dh ≤ 0,∀s ∈ S (17)

For n larger enough, we must have pol > 0,∀l ∈ L. Otherwise, every con-sumer would choose xh

ol = n and we would have contradicted (15) But whenpol > 0 we must have

∫

H

(xhol − ωh

ol + (Mhϕh)l)dh = 0 ∀l ∈ L (18)

since the aggregate budget constraint of the first period is a null sum of nonpositive terms and therefore a sum of null terms.¥

Asymptotics of truncated free-disposal equilibria

Now let (xhn, θ

hn, ϕh

n, (Mhn )h∈H), pn, π1n, qn, γn, Cn, Nn be the sequence

of free-disposal equilibria corresponding to En. Let n →∞ and examine theasymptotic properties of the sequence.

Lemma 2 pnsl 9 0 ∀s,l

Proof:Income in each state of the second period is the value of the bundle

ωhs + Ysxo (which is bounded away from zero in each coordinate) plus an

additional income equal to Nsjθ+psYsMϕ−∑j Dsjϕj ≥ 0. Since preferences

are time and state separable and monotonic, for any s and any l we havepn

sl 9 0. In fact, even in the presence of an unbounded increase in income,possibly offsetting the increase in xsl for an inferior good, the expenditure insome commodity would have to grow unboundedly and therefore ‖xhn

sl ‖ → ∞

22

for every h,implying that the feasibility equations would be violated for nsufficiently large. This completes the proof of this lemma.¥

The sequences Mhnjl n and Chn

jl nadmit (maxs,k,j Rjsk)/(mins pslYsl) as

an upper bound. In fact, any choice of collateral coefficients beyond thisbound determines sure repayment and would be equivalent to constitutingcollateral just up to this bound and consuming the remaining in the form ofa bundle not serving as collateral (that is, as part of xn

0 ).

Lemma 3 Cnj 6→ 0 as n → ∞. Actually, there exist uniform positive lower

bounds, across consumers, for the sequence Mhn of equilibrium collateral co-efficients

Proof:Let Shn

j = s ∈ S : pns R

js > pn

s YsMhnj be the set of states where agent

h gives default in promise j and let(Shn

j

)′be it’s complement. Now,

(Shn

j

)′6= ∅ ∀ h, j for n large enough, when asset j is traded. Other-

wise the Kuhn-Tucker first order condition in Mjl (which is necessary sincethe jacobian of the constraints with respect to (x0, x−0) has rank S) would

becomeu′ol

λoϕj ≤ 0, which is impossible.

Now let T nsj =

z ∈ Rl

+ : pns Ysz ≥ pn

s Rsj

and T n

j =S⋃

s=1

T nsj. Then, for

each n, ∀h, Mhnj ∈ T n

j and Cnj ∈ conT n

j . Notice that for n large enough

pnsl 6= 0 and therefore 0 /∈ conT n

j . Define the corresponding sets at the cluster

point (ps)Ss=1 À 0 : Tsj =

z ∈ IRl

+ : psYsz ≥ psRsj

and Tj =

S⋃s=1

Tsj. We

must have the cluster point Cj of the sequence Cnj belonging to conTj which

does not contain the origin, hence Cj 6= 0. This completes the proof of lemma3.¥

Lemma 4 ∫H

(xhn, ϕh

n,Mhnϕh

n)dh is a bounded sequence.

23

Proof:By definition of equilibrium,

∫H

xhnodh ≤ ∫

Hωh

o dh and∫

HMh

nϕhndh ≤ ∫

Hωh

o dh.

So ∫

H

xhnsdh <

∫

H

(ωhs + 2Ysω

ho )dh, ∀s ∈ S. (19)

For each l ∈ L the following holds∫

H

Mhlnjϕ

hndh = Clnj

∫

H

ϕhjndh (20)

and therefore

Cjnl

∫

H

ϕhnjdh ≤

∫

H

ωholdh, ∀l ∈ L (21)

Then, by lemma 3,∫

Hϕh

njdh is bounded. ¥

Lemma 5 The aggregate purchase of the derivative can also be taken asbounded, along the sequence of equilibria for the truncated economies.

Proof:

Let N(n) = maxs Nnsj and use the homogeneity of degree -1 of demand

for the derivative with respect to (Nj, π1j) to replace Nnj , πn

1j and θnhj by

Nnsj = Nn

sj/N(n), πn1j = πn

1j/N(n) and θhnj = θhn

j N(n). Then, Nnsj has a clus-

ter point also,∀s and actually, passing to a subsequence if necessary, Nnsj is

equal to one for some s and every n. Now,

Nsj

∫

H

θhj dh ≤

∫

H

minpsRsj, psYsM

hj

ϕh

j dh ≤ psRsj

∫

H

ϕhj dh

and therefore∫

Hθhn

j dh 9∞.

In the rest of the proof, to simplify the notation, let us take θn to beactually the allocation θn.¥

24

Lemma 6 the sequence of allocations xon, θn is uniformly bounded.

Proof:By the two preceding lemmas, the sequence zh

n ≡ (xhn, θh

n, ϕhn,M

hn ) satisfies

the hypothesis of the weak version of Fatou’s Lemma. Therefore ∃z integrablesuch that

zh ∈ clzn(h) for a.e h

Notice also that pn, π1n, qn, γn have cluster points.This implies that zh isbudget feasible at (p, q, π1, γ, C, N) = limn→∞(pn, qn, πn

1 , γn, Cn, Nn), passingto a subsequence if necessary . Moreover, (xh, θh, ϕh,Mh) maximizes Uh

at the cluster point of (pn, qn, γn, Cn, Nn), for almost every h. This is aconsequence of the fact that consumers’ optimal choice correspondences areclosed (see appendix).

Individual optimality at the cluster points implies that pn0l 9 0 (l =

1, ..., L)l and πn1j 9 0 (j = 1, ..., J). It follows immediately that

xhnol , θhn

j ≤ (ess suph,l

ωhol)/(min

l,j

lim

n→∞pn

ol, limn→∞

πn1j

). ¥

Lemma 7 The short sales allocation is also uniformly bounded

Proof:

Suppose not, then there is a sequence h(n) of agents such that ϕh(n)n →∞,

even though ϕhn 9∞ for almost every h ∈ h(n)n. Now:

pnoM

h(n)j − πh(n)2j = αn

j + (pno −

∑s

γns pn

s Ys)Cnj

where

αnj = −qn

j +∑

s

γns (pn

s Rsj − pns YsM

h(n)j )+ +

∑s

γns pn

s YsMh(n)j

Claim ϕh(n)n →∞ =⇒ αn

j − (−qnj +

∑s γn

s pns Rsj) → 0

From first order conditions in xns , we have:

u′s − λspns = 0 (22)

25

Now Mh(n)j ϕ

h(n)j → ∞ implies pn

s YsMh(n)j ϕ

h(n)j → ∞. In a nondefault state,

we have (u′s)h(n) → 0. In fact, let yn → y and h be a cluster point of h(n),

then limn(u′s)h(n)(yn) = limn(u′s)

h(yn) = 0, by assumption (P) and Moore’s

lemma (see Dunford-Schwartz [11] I.7.6). Then in (22), we obtain λh(n)s → 0.

Now λh(n)o 9 0, since first period wealth pn

oωo 9 ∞ (recall short salesinduce a net cost).

Then, we have that:

(u′oλo

)h(n)

→ 0,

again, by assumption (P). Now let

S1 = s : pns Rsj > pn

s YsMh(n)j

S2 = s : pns Rsj < pn

s YsMh(n)j

S3 = s : pns Rsj = pn

s YsMh(n)j

and let ψ(Mj) = minpsYsMj, psRsj. Since the jacobian of the budgetconstraints with respect to x is of rank S, the first order condition on Mj isnecessary and can be written as

(u′oλo

)h(n)

−∑s∈S2

γns pn

s Ys+∑s∈S2

(λs

λo

)h(n)

pns Ys−

∑s∈S3

γns νh(n)+

∑s∈S3

(λs

λo

)h(n)

νh(n) = 0

where νh(n) ∈ ∂Mjψ = conlim∇ψ(zi) : zi → Mj, zi ∈ dom (∇ψ\O),

where O is a zero measure set (see Clarke [6], 2.5.1 and 6.1.1).

Now(

u′oλo

)h(n)

,(

λs

λo

)h(n)

s=1,..,S→ 0. Then we must have γn

s → 0 for s ∈ S2.

But also we have

αnj = −qn

j +∑

s∈S1⋃

S3

γns pn

s Rsj +∑s∈S2

γns pn

s YsMh(n)j

26

and thereforeαn

j

−qnj +

∑s γn

s pns Rsj

→ 1

So the claim is established.2

Then αj does not depend on Mj in the limit.

Now (pnoM

h(n)j − π

2h(n)j )ϕ

h(n)j bounded implies (pn

oMh(n)j − π

2h(n)j ) → 0,

hence

−qnj +

∑s

γns pn

s Rsj + (pno −

∑s

γns pn

s Ys)Cnj → 0

That is qj =∑

s γspsRsj + (po −∑

s γspsYs)Cj in the limit, but qj mustbe less than

∑s γspsRsj + (po −

∑s γspsYs)Cj. Otherwise any agent could

make poMhj − π2h

j = 0 for Mhj sufficient small but different from zero so that

default occurs in every state. Such a choice for Mhj would be accompanied

by choosing ϕj arbitrary large, which can not occur since there is a finite op-timal choice znh for almost every h (see the argument in the proof of lemma6). ¥

Then, Mhjlnϕh

jn is uniformly bounded and, from (2), xhsln is also uni-

formly bounded. All these facts imply that the sequence (xn, θn, ϕn,Mnϕn)is uniformly bounded.

We can now continue the proof of existence of equilibria for the economyE using the strong version of Fatou’s lemma (see Appendix):

∫H

xhdh = limn→∞∫

Hxh

ndh,∫

Hθhdh = limn→∞

∫H

θhndh,

∫H

ϕhdh = limn→∞∫

Hϕh

ndh and

∫

H

Mhϕhdh = limn→∞

∫

H

Mhnϕh

ndh

Thus all markets clear in the E . We also have Cjl

∫H

ϕhj dh =

∫H

Mhjlϕ

hj dh.

27

Moreover, Nsj

∫H

θhj dh =

∫H

minpsRsj, psYsM

hj

ϕh

j dh. Suppose not,then, using the notation in the proof of lemma 5, we would have for all nlarge enough

∫H

Dhnj ϕhn

j dh/(N(n)∫

Hθhn

j dh) > n , for some (s, j), implying∫H

Dhnj ϕhn

j dh/∫

Hθhn

j dh > nN(n). If N(n) < n, then the inequality wouldhold as equality. If N(n) = n, then n2

∫H

θhnj dh would be bounded, implying

that∫

Hθhn

j dh = n∫

Hθhn

j dh → 0. Now, πn1j

∫H

θhnj dh =

∑s γn

s

∫H

Dhnsj ϕhn

j dhwhere πn

1j = πn1j/N(n) is bounded. Hence γn → 0 or

∫H

Dhnj ϕhn

j dh → 0, butthe former implies the latter, since we would have πh

2j = p0(Mj − Cj) whichwould lead every agent to choose Mh

j ≥ Cj,∀h, and, therefore, Mhj = Cj,

implying that πh2j = 0 and ϕh

j = 0,∀h, contradicting the supposed strictinequality.

Moreover, π1j =∑

s γsNsj when asset j is traded, since

π1j

∫

H

θhj dh =

∫

H

πh2jϕ

hj dh =

∑s

γs

∫

H

Dhsjϕ

hj dh =

∑s

γsNsj

∫

H

θhj dh.

7 Efficency

In this section we prove that an equilibrium allocation is constrained effi-cient among all feasible allocations that provide income across states throughthe same spot prices (the given equilibrium prices). In comparison with theequilibrium obtained by [3], we can say that our equilibrium is Pareto supe-rior, since we are not impossing any kind of bounded short sale.

As in the work of Magill and Shafer [18], we compare the equilibriumallocation with one feasible allocation whose portfolios do not necessarilyresult from trading competitively in asset markets. That is, in alternativeallocations agents pay participation fees which may differ from the marketportfolio cost. Equivalently, we allow for transfers across agents which arebeing added to the usual market portfolio cost.

Proposition 1 Let ((x, θ, ϕ, M), p, π1, π2, C, N) be an equilibrium. The al-location (x, θ, ϕ, M) is efficient among all allocations (x, θ, ϕ, M) for whichthere are transfers T h ∈ IR across agents and a vector C ∈ IRJL

+ , such that

(i)∫

H(xh

o + Mhϕh)dh =∫

Hωh

o dh,∫

Hxh

s =∫

H(ωh

s + YsMhϕh + Ysx

ho)dh,

π1

∫H

θhdh =∫

Hπ2ϕ

hdh

28

(ii)

ps(xhs − ωh

s − Ysxho) +

∑j∈J

minpsRjs, psYsM

hj ϕh

j

=∑j∈J

Nj

sθhj +

∑j∈J

psYsMhj ϕh

j , ∀s, a.e. h

(iii) po(xho + Mhϕh − ωh

o ) + π1θh − π2ϕ

h + T h = 0

(iv)∫

HT hdh = 0

(v) Cj

∫H

ϕhj dh =

∫H

Mhj ϕh

j , ∀jwhere the equilibrium prices are given by

π1 = q −∑

s

γsg1s

and

π2 = q −∑

s

γsg2s + (po −∑

s

γspsYs)M j − (po −∑

s

γspsYs)Cj

Proof:Suppose not, say (x, θ, ϕ, M,C) together with some transfer fraction T

satisfies (i) through (v); uh(xho +Mhϕh, xh

−o) ≥ uh(xho +M

hϕh, x−o) for a.e h

and uh(xho +Mhϕh, xh

−o) > uh(xho +M

hϕh, x−o) for h in some positive measure

set G of agents. Then, for h ∈ G, the first period constraint must be violated,that is,

po(xho + Mhϕh − ωh

o ) + π1θh − π2ϕ

h > 0 (23)

Now remember that

ghs = (psRs − psYsM

h)+ϕh − (psRs −Ns)+θh

= (psRs −Dhs )ϕh − (psRs −N s)θ

h

By continuity of preferences and monotonicity we can take G = H, withoutloss of generality. Then

∫H

ghs dh > 0 for some s, by (23) and (i), implying∫

HNsθ

hdh >∫

HDh

s ϕhdh. Now, by (ii),

ps.

∫

H

(xhs − ωh

s − Ys(Mhϕh + xh

o))dh =

∫

H

Nsθhdh−

∫

H

Dhs ϕhdh

29

where the right hand side is strictly positive, contradicting∫H

(xhs − ωh

s − Ys(Mhϕh + xh

o))dh = 0 .¥

The above weak constrained efficiency property is in the same spirit asproperties found in the incomplete markets model without default (see [18])and also in the exogenous collateral model (without utility penalties) of [9].As in these models, it does not seem to be possible to show that equilib-rium allocations are undominated when prices are no longer assumed to beconstant at the equilibrium levels. However equilibria with default and en-dogenous collateral, as proposed in this paper, is Pareto superior to theno-default equilibria, to the exogenous collateral equilibria and even to thebounded short-sales endogenous collateral equilibria of [3], since our equilib-ria is free of any of the constraints which are used in the definition of theseequilibrium concepts (that is, absence of default, exogeneity of collateral andbounded short-sales).

8 Conclusions

In this paper we have obtained a no arbitrage characterization of theprices of collateralized promises, where the collateral coefficients are choosenby borrowers as in [3]. We also obtained a pricing result consistent with theobservation made by [16] for the case of short sale constraints, more preciselywe have shown that our buy and net sell prices are supermartingale and sub-martingales, respectively, under some probability measures. For these prob-abilities we have found lower and upper bounds for the prices of derivativeswritten on the primitive defaultable assets. Finally using the nonarbitragecharacterization of asset prices we proposed an equilibrium pricing formulaand showed the existence of equilibrium in the model where borrowers choosethe collateral coefficients, without imposing uniform bounds on short-sales(thus avoiding a drawback of the work by [3]) and showed that this equilib-rium is constrained efficient.

30

9 Appendix

9.1 Mathematical Preliminarities

• Let C(K) the Banach space of continuous functions on the compactmetric space K. Let L1(H, C(K)) be the Banach space of Bochner in-tegrable functions whose values belong to C(K). For z ∈ L1(H,C(K)),

||z||1 :=

∫

H

supK|Zh|dh < ∞

Let B(K) denotes the set of regular measures on the Borelians of K.The dual space of L1(H, C(K)) is L∞ω (H,B(K)), the Banach spaceof essentially strong bounded weak ∗ measurable functions from Hinto B(K). We say that µn ⊂ L∞(H,B(K)) converges to µ ∈L∞ω (H,B(K)) with respect to the weak * topology on the dual L1(H, C(K)),if ∫

H

∫

K

zhdµhndh →

∫

H

∫

K

zhdµhdh , ∀f ∈ L1(H,C(K))

• We will use in this work the following lemmas ( in m-dimension).Fatou’s lemma (Weak Version)Let fn be a sequence of integrable functions of a measure space(Ω,A, ν) into IRm

+ . Suppose that limn→∞∫Ω

fndν exists. Then thereexists an integrable function f : Ω 7→ IRm

+ such that:

1. f(w) ∈ clfn(w) for a.e w, and

2.∫

Ωfdν ≤ limn→∞

∫Ω

fndν

Fatou’s lemma (Strong version)If in addition the sequence fn above is uniformly integrable, then theinequality in 2. holds as an equality.

9.2 Extended Game

We extend the generalized game by allowing for mixed strategies both inportfolios and collateral bundles. Remember that, for each player a mixedstrategy is a probability distribution on his set of pure strategies. In this

31

case the set of measures on the Borelians of Kn = [0, n]J × [0, n]J × [0, n]LJ .We denote by B the set of mixed strategies of each consumer. Since we arenot interested in a mixed strategies equilibrium, per se, we will extend theprevious game to a game J n over mixed strategies ( that we call extendedgame) whose equilibria: 1) exist 2) can be purified and 3) a pure version isan equilibrium for the original game. First, before extending the game tomixed strategies, let us rewrite the payoffs of the fictitious agents replacingconsumption bundles by the following function of portfolios and collateral:

dh(θh, ϕh,Mh) = arg maxuh : xh ∈ [0, n]L(S+1) satisfies (1) and (2)

That is, function dh solves the utility maximization problem for a givenportfolio (θh, ϕh) and given collateral coefficients Mh

j . By the maximumtheorem and the fact that consumers’ choice correspondences are closed (seeProposition below), dh is continuous. Secondly, we extend the payoffs tomixed strategies.

(i) Each consumer h ∈ H chooses (xh, µh) ∈ [0, n]L(S+1) × B in order tomaximize

∫Kn

Uh(xho +Mhϕh, xh

−o)dµh subject to the following extendedbudget constraints:

po(xho − ωh

o ) +

∫

Kn

[π1θh + poM

hϕh − πh2ϕh]dµh ≤ 0

ps(xhs −ωh

s −Ysxho) ≤

∫

Kn

∑j

(N js θ

hj −Dj

sϕhj + psYsM

hj ϕh

j )dµh for s ∈ S

(ii) The auctioneer of the first period chooses po ∈ 4L−1 in order to maxi-mize

po

∫

H

∫

Kn

[dho(θ

h, ϕh,Mh) +∑

j

Mhj ϕh

j − ωho ]dµhdh

(iii) The auctioneer of state s in the second period chooses ps ∈ 4L−1 inorder to maximize

ps

∫

H

∫

Kn

[dhs (θ

h, ϕh,Mh)−∑

j

YsMhj ϕh

j −ωhs − Ysd

ho(θ

h, ϕh,Mh)]dµhdh

32

(iv) The first JL fictitious agents chooses Cjl ∈ [0, n] in order to minimize

(

∫

H

∫

Kn

[Cjlθhj −Mh

jlϕhjl]dµhdh)2

• Another fictitious agent chooses π1j ∈ [0, 1] , qj ∈ [0, maxs,k Rsjk] , Nsj ∈[0, n] and γs ∈ [0, γ] for every j and s in order to minimize

∑j

((π1j

∫

H

∫

Kn

θhj dµhdh−

∫

H

∫

Kn

πh2jϕ

hj dµhdh)2

+(qj −∑

s

γspsRsj)2

∫

H

∫

Kn

θhj dµhdh

+∑

s

(Nsj

∫

H

∫

Kn

θhj dµhdh−

∫

H

∫

Kn

minpsRsj, psYsM

hj

dµhdh)2

)

Lemma 8 J n has an equilibrium, possibly in mixed strategies over portfolioand collateral together.

Proof:The existence argument in Ali Khan [17] can be modified to allow for some

atomic players. First, by the Proposition below, consumers’ pure strategieschoice correspondences are closed, and therefore, upper semicontinuous inthe truncated economy. Now, mixed strategies choice correspondences arethe closed convex hull of the pure strategies choice correspondences and,therefore, will be also upper semicontinuous.

Now, define the correspondence:

α(p, π, C) = f ≡ (x, µ) ∈ ([0, n]L(S+1) × B)H : f(h) ∈ νh(p, π, C)

Which is also convex valued and upper semicontinuous . The best re-sponse correspondences Ri of the r = S + 2 + JL fictitious agents are con-vex valued and upper semicontinuous on the profile of consumers’ proba-bility measures on Kn (with respect to the weak * topology on the dual ofL1(H, C(Kn) ). The profiles set is compact for the same topology and Fan -Glicksberg fixed point theorem applies to α×∏r

i=1Ri. ¥

33

Lemma 9 J n has an equilibrium in pure strategies.

Proof:In this part Liapunov’s theorem will be fundamental. First, notice that

the payoffs of the atomic players in J n depend on the profile of mixedstrategies (µh)h only through finitely many e indicators of the form (e =L + S + SL + 2JL).

∫

H

∫

Kn

Zhe (θh, ϕh,Mh)dµhdh where Ze ∈ L(H,C(Kn))

Secondly, let Eh(p, π, C) =∏

2 νh(p, π, C) and Z = (Z1, . . . , Ze). Now,

∫

Kn

Zh(θh, ϕh,Mh)dEh(p, π, C) = conv

∫

Kn

Zh(θh, ϕh,Mh)d(extEh(p, π, C))

where the integral on the left hand side is the set in IRe of the all integralsof the form

∫Kn

Zh(θh, ϕh,Mh)dµh, for µh ∈ Eh(p, π, C). The integral onthe right hand side is defined endogenously. The equality above follows bylinearity of the map

µh 7→∫

Kn

Zh(θh, ϕh,Mh)dµh

Then, Theorem I.D.4 in Hildenbrand [15] implies

∫

H

∫

Kn

Zh(·)dEh(p, π, C)dh =

∫

H

∫

Kn

Zh(·)d(extEh(p, π, C))dh

Then, given a mixed strategies equilibrium profile (µh)h, there exists(θh, ϕh,Mh) such that the Dirac measure at (θh, ϕh,Mh) is an extreme pointof Eh (evaluated at the equilibrium levels of the variables chosen by theatomic players ) and (θh, ϕh,Mh)h can replace (µh)h and keep all equilibriumconditions satisfied, without changing the equilibrium levels of the variableschosen by the atomic players but replacing the former equilibrium bundlesby dh(θh, ϕh, Mh) .¥

34

9.3 Closedness of consumers’ choice correspondences

Since p0 ∈ 4L−1 consumers’ budget correspondence always has the origin asan interior point of its values, implying that the interior of the budget corre-spondence is lower-semicontinuous and, therefore, the budget correspondenceitself is also lower semi-continuous.

Lemma 10 The budget correspondence is lower semicontinuous .

Proof:Define Bh

o (p, q, γ, C) to be the interior of Bh(p, q, γ, C) and let xh = 0,θh = 0, ϕh = 0 and Mh = 0. The values thus chosen for these variables satisfythe budget constraint of agent h with strict inequality. So, Bh

o (p, q, γ, C) 6= φ.

Let limk→∞(pk, qk, γk, Ck) = (p, q, γ, C) and (xh, θh, ϕh,Mh) ∈ Bho (p, q, γ, M).

Then for every (xhk, θ

hk , ϕh

k,Mhk ) such that

limn→∞

(xhk, θ

hk , ϕh

k,Mhk ) = (xh, θh, ϕh,Mh)

and for n large enough, the strict budget inequalitis hold. Thus

(xhk, θ

hk , ϕh

k, mhk) ∈ Bh

o (pk, qk, γk, Ck)

for k large enough, which implies that Bho is lower hemi-continuous. Then

the result follows from [15], pag. 26, fact 4.

It is immediate to see that budget correspondences of truncated economiesenjoy also the same property. Let us see that choice correspondences oftruncated economies are closed. Consumers’ optimal choice correspondencesare closed at any (p, q, γ, C) satisfying the assumptions of the previous lemma:if (pk, qk, γk, Ck) → (p, q, γ, C), zk is an optimal choice of consumer h at(pk, qk, γk, Ck) and zk → z, given any z ∈ Bh(p, q, γ, C), ∃(zk) → z such thatzk ∈ Bh(pk, qk, γk, Ck) and zk is not prefered to zk by consumer h, implying,by continuity of uh that z is an optimal choice at (p, q, γ, C).

Comment

Consider an economy where derivative and primitive asset aggregates arealso required to match in value ( but not in quantity) but collateral margin

35

requirements are bounded from below, say p0Mhj − π2j ≥ ε (or that p0M

hj −

qj ≥ ε, as in [3]), when ϕhj > 0. Then, the lower semi-continuity of the

budget correspondence holds. In fact, taking p0 ∈ 4L−1 and πh2j ≡ qj −∑

s γs(psRsj − psYsMj)+ (that is, setting the negative tail in the equilibrium

sale price of primitive assets to be δj ≡ −(p0−∑

s γspsYs)Mhj ), the constraint

p0Mhj − qj ≥ ε is always well-defined and admits an interior solution (with

Mhj large enough) which is compatible with the interior solution (x, θ, ϕ) = 0

of the other budget constraints.

References

[1] Araujo A., P. Monteiro and M. Pascoa, (1998), “ Infinte Horizon Incom-plete Markets with a Continuum of States”, Mathematical Finance. 6,119–132.

[2] Araujo A., P. Monteiro and M. Pascoa, (1998), “ Incomplete Markets,Continuum of states and Default”, Economic Theory. 11, 205–213.

[3] Araujo A., J. Orrillo and M. Pascoa, (2000), “ Equilibrium with Defaultand Endogenous Collateral”, Mathematical Finance. 10, No 1, 1–21.

[4] Araujo A., and M. Pascoa, (2002), “ Bankruptcy in a Model of Unse-cured Claims”, Economic Theory. 20, 455–481.

[5] Araujo A.,M. Pascoa, and J. Torrez-Martinez (2002), “ Collateral AvoidsPonzi Schemes in Incomplete Markets”, Econometrica. 70 (4).1613–1638

[6] Clarke, F. (1983), Optimization and Nonsmooth Analysis. Wiley andSons.

[7] Cvitanic, J. and I. Karatzas, (1993), “ Hedging Contingent Claims withConstrained Portfolios”, Annals of Applied Probability 3, 652–681.

[8] Dubey, P., J.Geanakoplos and M.Shubik, (1989), “Liquidity andbankruptcy with incomplete markets: Pure exchange”. Cowels Foun-dation discussion paper 900.

36

[9] Dubey, P.,J.Geanakoplos and W. Zame, (1995), “Default, Collateral,and Derivatives”. Yale University, mimeo.

[10] Duffie, D., and W. Shafer, (1985), “Equilibrium in Incomplete MarketsI: A Basic Model of Generic Existence ”, Journal of Mathematical Eco-nomics 14, 285–300.

[11] Dunford, N., and Schwartz, J. (1958), Linear Operators. IntersciencePublisher.

[12] El Karoui, N. and M. Quenez, (1995), “Dynamic Programming andPricing of Contingent Claims in an Incomplete Market”, SIAM Journalof Control and Optimization 33, 29–66.

[13] Fajardo, J., (2002), “ A Note on Arbitrage and Exogenous Collateral”.Working Paper IBMEC.

[14] Hart, O., (1975), “On the Optimality of Equilibrium when the MarketStructure is Incomplete”, Journal of Economic Theory 11, 418–430.

[15] Hildebrand , W., (1974), Core and Equilibria of a Large Economy,Princeton University Press, Princeton.

[16] Jouini,E. and H. Kallal, (1995),“Arbitrage in security markets withshort-sales constraints”, Mathematical Finance 5(3),197–232.

[17] Khan, M.A., (1986), “Equilibrium points of nonatomic games over aBanach”,Transactions of the American Mathematical Society 293, n.2.

[18] Magill,M. and W. Shafer, (1991), “ Incomplete Markets”, in W. Hin-denbrand and H. Sonnenschein, Handbook of Mathematical economics,Vol. IV, North Holland, Amsterdam.

37

ENSAIOS ECONÔMICOS DA EPGE 466. INFLAÇÃO E FLEXIBILIDADE SALARIAL - Marcelo Côrtes Neri; Maurício Pinheiro –

Dezembro de 2002 – 16 págs.

467. DISTRIBUTIVE EFFECTTS OF BRAZILIAN STRUCTURAL REFORMS - Marcelo Côrtes Neri; José Márcio Camargo – Dezembro de 2002 – 38 págs.

468. O TEMPO DAS CRIANÇAS - Marcelo Côrtes Neri; Daniela Costa – Dezembro de 2002 – 14 págs.

469. EMPLOYMENT AND PRODUCTIVITY IN BRAZIL IN THE NINETIES - José Márcio Camargo; Marcelo Côrtes Neri; Maurício Cortez Reis – Dezembro de 2002 – 32 págs.

470. THE ALIASING EFFECT, THE FEJER KERNEL AND TEMPORALLY AGGREGATED LONG MEMORY PROCESSES - Leonardo R. Souza – Janeiro de 2003 – 32 págs.

471. CUSTO DE CICLO ECONÔMICO NO BRASIL EM UM MODELO COM RESTRIÇÃO A CRÉDITO - Bárbara Vasconcelos Boavista da Cunha; Pedro Cavalcanti Ferreira – Janeiro de 2003 – 21 págs.

472. THE COSTS OF EDUCATION, LONGEVITY AND THE POVERTY OF NATIONS - Pedro Cavalcanti Ferreira; Samuel de Abreu Pessoa – Janeiro de 2003 – 31 págs.

473. A GENERALIZATION OF JUDD’S METHOD OF OUT-STEADY-STATE COMPARISONS IN PERFECT FORESIGHT MODELS - Paulo Barelli; Samuel de Abreu Pessoa – Fevereiro de 2003 – 7 págs.

474. AS LEIS DA FALÊNCIA: UMA ABORDAGEM ECONÔMICA - Aloísio Pessoa de Araújo – Fevereiro de 2003 – 25 págs.

475. THE LONG-RUN ECONOMIC IMPACT OF AIDS - Pedro Cavalcanti G. Ferreira; Samuel de Abreu Pessoa – Fevereiro de 2003 – 30 págs.

476. A MONETARY MECHANISM FOR SHARING CAPITAL: DIAMOND AND DYBVIG MEET KIYOTAKI AND WRIGHT – Ricardo de O. Cavalcanti – Fevereiro de 2003 – 16 págs.

477. INADA CONDITIONS IMPLY THAT PRODUCTION FUNCTION MUST BE ASYMPTOTICALLY COBB-DOUGLAS - Paulo Barelli; Samuel de Abreu Pessoa – Março de 2003 – 4 págs.

478. TEMPORAL AGGREGATION AND BANDWIDTH SELECTION IN ESTIMATING LONG MEMORY - Leonardo R. Souza - Março de 2003 – 19 págs.

479. A NOTE ON COLE AND STOCKMAN - Paulo Barelli; Samuel de Abreu Pessoa – Abril de 2003 – 8 págs.

480. A HIPÓTESE DAS EXPECTATIVAS NA ESTRUTURA A TERMO DE JUROS NO BRASIL: UMA APLICAÇÃO DE MODELOS DE VALOR PRESENTE - Alexandre Maia Correia Lima; João Victor Issler – Maio de 2003 – 30 págs.

481. ON THE WELFARE COSTS OF BUSINESS CYCLES IN THE 20TH CENTURY - João Victor Issler; Afonso Arinos de Mello Franco; Osmani Teixeira de Carvalho Guillén – Maio de 2003 – 29 págs.

482. RETORNOS ANORMAIS E ESTRATÉGIAS CONTRÁRIAS - Marco Antonio Bonomo; Ivana Dall’Agnol – Junho de 2003 – 27 págs.

483. EVOLUÇÃO DA PRODUTIVIDADE TOTAL DOS FATORES NA ECONOMIA BRASILEIRA: UMA ANÁLISE COMPARATIVA - Victor Gomes; Samuel de Abreu Pessoa;Fernando A . Veloso – Junho de 2003 – 45 págs.

484. MIGRAÇÃO, SELEÇÃO E DIFERENÇAS REGIONAIS DE RENDA NO BRASIL - Enestor da Rosa dos Santos Junior; Naércio Menezes Filho; Pedro Cavalcanti Ferreira – Junho de 2003 – 23 págs.

485. THE RISK PREMIUM ON BRAZILIAN GOVERNMENT DEBT, 1996-2002 - André Soares Loureiro; Fernando de Holanda Barbosa - Junho de 2003 – 16 págs.

486. FORECASTING ELECTRICITY DEMAND USING GENERALIZED LONG MEMORY - Lacir Jorge Soares; Leonardo Rocha Souza – Junho de 2003 – 22 págs.

487. USING IRREGULARLY SPACED RETURNS TO ESTIMATE MULTI-FACTOR MODELS: APPLICATION TO BRAZILIAN EQUITY DATA - Álvaro Veiga; Leonardo Rocha Souza – Junho de 2003 – 26 págs.

488. BOUNDS FOR THE PROBABILITY DISTRIBUTION FUNCTION OF THE LINEAR ACD PROCESS – Marcelo Fernandes – Julho de 2003 – 10 págs.

489. CONVEX COMBINATIONS OF LONG MEMORY ESTIMATES FROM DIFFERENT SAMPLING RATES - Leonardo R. Souza; Jeremy Smith; Reinaldo C. Souza – Julho de 2003 – 20 págs.

490. IDADE, INCAPACIDADE E A INFLAÇÃO DO NÚMERO DE PESSOAS COM DEFICIÊNCIA - Marcelo Neri ; Wagner Soares – Julho de 2003 – 54 págs.

491. FORECASTING ELECTRICITY LOAD DEMAND: ANALYSIS OF THE 2001 RATIONING PERIOD IN BRAZIL - Leonardo Rocha Souza; Lacir Jorge Soares – Julho de 2003 – 27 págs.

492. THE MISSING LINK: USING THE NBER RECESSION INDICATOR TO CONSTRUCT COINCIDENT AND LEADING INDICES OF ECONOMIC ACTIVITY - JoãoVictor Issler; Farshid Vahid – Agosto de 2003 – 26 págs.

493. REAL EXCHANGE RATE MISALIGNMENTS - Maria Cristina T. Terra; Frederico Estrella Carneiro Valladares – Agosto de 2003 – 26 págs.

494. ELASTICITY OF SUBSTITUTION BETWEEN CAPITAL AND LABOR: A PANEL DATA APPROACH - Samuel de Abreu Pessoa ; Silvia Matos Pessoa; Rafael Rob – Agosto de 2003 – 30 págs.

495. A EXPERIÊNCIA DE CRESCIMENTO DAS ECONOMIAS DE MERCADO NOS ÚLTIMOS 40 ANOS – Samuel de Abreu Pessoa – Agosto de 2003 – 22 págs.

496. NORMALITY UNDER UNCERTAINTY – Carlos Eugênio E. da Costa – Setembro de 2003 – 08 págs.

497. RISK SHARING AND THE HOUSEHOLD COLLECTIVE MODEL - Carlos Eugênio E. da Costa – Setembro de 2003 – 15 págs.

498. REDISTRIBUTION WITH UNOBSERVED 'EX-ANTE' CHOICES - Carlos Eugênio E. da Costa – Setembro de 2003 – 30 págs.

499. OPTIMAL TAXATION WITH GRADUAL LEARNING OF TYPES - Carlos Eugênio E. da Costa – Setembro de 2003 – 26 págs.

500. AVALIANDO PESQUISADORES E DEPARTAMENTOS DE ECONOMIA NO BRASIL A PARTIR DE CITAÇÕES INTERNACIONAIS - João Victor Issler; Rachel Couto Ferreira – Setembro de 2003 – 29 págs.

501. A FAMILY OF AUTOREGRESSIVE CONDITIONAL DURATION MODELS - Marcelo Fernandes; Joachim Grammig – Setembro de 2003 – 37 págs.

502. NONPARAMETRIC SPECIFICATION TESTS FOR CONDITIONAL DURATION MODELS - Marcelo Fernandes; Joachim Grammig – Setembro de 2003 – 42 págs.

503. A NOTE ON CHAMBERS’S “LONG MEMORY AND AGGREGATION IN MACROECONOMIC TIME SERIES” – Leonardo Rocha Souza – Setembro de 2003 – 11págs.

504. ON CHOICE OF TECHNIQUE IN THE ROBINSON-SOLOW-SRINIVASAN MODEL - M. Ali Khan – Setembro de 2003 – 34 págs.

505. ENDOGENOUS TIME-DEPENDENT RULES AND THE COSTS OF DISINFLATION WITH IMPERFECT CREDIBILITY - Marco Bonomo; Carlos Viana de Carvalho – Outubro de 2003 – 27 págs.

506. CAPITAIS INTERNACIONAIS: COMPLEMENTARES OU SUBSTITUTOS? - Carlos Hamilton V. Araújo; Renato G. Flôres Jr. – Outubro de 2003 – 24 págs.

507. TESTING PRODUCTION FUNCTIONS USED IN EMPIRICAL GROWTH STUDIES - Pedro Cavalcanti Ferreira; João Victor Issler; Samuel de Abreu Pessoa – Outubro de 2003 – 8 págs.

508. SHOULD EDUCATIONAL POLICIES BE REGRESSIVE ? Daniel Gottlieb; Humberto Moreira – Outubro de 2003 – 25 págs.

509. TRADE AND CO-OPERATION IN THE EU-MERCOSUL FREE TRADE AGREEMENT - Renato G. Flôres Jr. – Outubro de 2003 – 33 págs.

510. OUTPUT CONVERGENCE IN MERCOSUR: MULTIVARIATE TIME SERIES EVIDENCE - Mariam Camarero; Renato G. Flôres Jr; Cecílio Tamarit – Outubro de 2003 – 36 págs.

511. ENDOGENOUS COLLATERAL - Aloísio Araújo; José Fajardo Barbachan; Mario R. Páscoa – Novembro de 2003 – 37 págs.

512. NON-MONOTONE INSURANCE CONTRACTS AND THEIR EMPIRICAL CONSEQUENCES - Aloísio Araujo; Humberto Moreira – Novembro de 2003 – 31 págs.

513. EQUILIBRIA IN SECURITY MARKETS WITH A CONTINUUM OF AGENTS - A. Araujo; V. F. Martins da Rocha; P. K. Monteiro – Novembro de 2003 – 17 págs.

514. SPECULATIVE ATTACKS ON DEBTS AND OPTIMUM CURRENCY AREA: A WELFARE ANALYSIS - Aloisio Araujo; Márcia Leon – Novembro de 2003 – 50 págs.